Does the stool test for Helicobacter pylori show only acute infections?

There are four options when being tested for Helicobacter pylori – blood test, stool test, breath test and scope test. (Helicobacter pylori (H. pylori) infection)

Scope test is definitely the most accurate one because the doctor can take a tissue sample from the stomach with it. Blood test is considered the least accurate because it can show antigens which could be the result of a previous infection with H. pylori and not an acute infection. (Which test is best for Helicobacter pylori? A cost-effectiveness model using decision analysis) Stool test looks for antigens as well but it seems to be positive only in patients with an acute infection. (The stool antigen test for detection of Helicobacter pylori after eradication therapy)

Does the stool test for Helicobacter pylori really show only acute infections or can it also be false positive if someone has already had an H. pylori infection? If it can be false positive, in how many cases is it false positive?